Recently, such approximation was shown to be in the class of Lévy
operators for a certain finite measure [1]}, [2]}, [3]}, [4]}. We thus present a unified approach to
numerical discretizations for local and nonlocal operators. In fact,
by choosing the correct weights, we recover either local, nonlocal, or
combinations of both. Hence, consider the family of bounded,
symmetric, and monotone operators \(\mathcal {L}^h: \ell ^1(\mathcal {G}_h)\rightarrow \mathbb {R}\)  given
by
\(\mathcal {L}^h[\psi ](x_\beta )=\mathcal {L}^h[\psi ]_\beta =\sum _{\gamma \ne 0} (\psi (x_\beta +z_\gamma )-\psi (x_\beta )) \omega _{\gamma ,h} \quad \text{for all $\psi :\mathcal {G}_h\rightarrow \mathbb {R}$},\) 
